The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Abstract

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom K (A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the N-graded second Hochschild homology group HH 2 (A) in the sense of Sköldberg.2010 Mathematics Subject Classification. 16E40, 16G20, 16L60.

An application of a theorem of Sheila Brenner for Hochschild extension algebras of a truncated quiver algebra

“…In this section, we recall theorems of Brenner [2], the ordinary quiver of a Hochschild extension algebra along [5] and the definition of an elementary cycle in the ordinary quiver of a Hochschild extension algebra introduced in [3]. After that we define an α-revived cycle for a 2-cocycle α and we give an example of these cycles.…”

“…We denote by D(A) the standard duality module Hom K (A, K). We recall the definitions of a Hochschild extension and a Hochschild extension algebra from [4], [5] and [10]. By a Hochschild extension over A by D(A), we mean an exact sequence…”

“…This paper is organized as follows: In Section 2, we recall results of Brenner, some definitions and facts about Hochschild extension algebras. In particular, some facts about 2-cocycles and the ordinary quiver of Hochschild extension algebras are based on [5] and [6]. In Section 3, we will give a characterization of nonzero oriented cycles in a Hochschild extension algebra.…”

“…For general facts on quivers and bound quiver algebras, we refer to [1] and [8]. Also for Hochschild extension algebra, we refer to [5], [6], [9] and [10]. Moreover, the notation including ∆ 0 , ∆ 1 , ∆ + and the isomorphism Θ : q D(HH 2, q (A))…”